If $\frac{x+1}{(x-1)(x-2)(x-3)}=\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{x-3},$then $A+C=$

$x+1=A(x-2)(x-3)+B(x-1)(x-3)+C(x-1)(x-2)$

$\begin{array}{l}putx=1\\ 2=2A\\ A=1\\ putx=3\\ 4=0+2C\\ C=2\end{array}$

$\begin{array}{l}A+C=1+2=3\\ putx=2\\ 3=0-B+0\\ B=-3\\ A+C=3=-B\end{array}$

$\begin{array}{l}{B}^2-12=9-12=-3\left(A\right)\\ 2B+C=2(-3)+2\\ =-6+2=-4\end{array}$